Good Gradings on Upper Block Triangular Matrix Algebras

We investigate group gradings of upper block triangular matrix algebras over a field such that all the matrix units lying there are homogeneous elements. We describe these gradings as endomorphism algebras of graded flags and classify them as orbits of a certain biaction of a Young subgroup and the group G on the set G ⁿ , where G is the grading group and n is the size of the matrix algebra. In particular, the results apply to algebras of upper triangular matrices.     

Identities and isomorphisms of graded simple algebras

Let G be an arbitrary abelian group and let A and B be two finite dimensional G-graded simple algebras over an algebraically closed field F such that the orders of all finite subgroups of G are invertible in F. We prove that A and B are isomorphic if and only if they satisfy the same G-graded identities. We also describe all isomorphism classes of finite dimensional G-graded simple algebras.     

Some Homological Properties of Skew PBW Extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well-known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.     

Hopf Algebras on Schurian Quivers

Let H be a Hopf algebra with a finite-dimensional, nontrivial space of skew primitive elements, over an algebraically closed field of characteristic zero. We prove that H contains either the polynomial algebra as a Hopf subalgebra, or a certain Schurian simple-pointed Hopf subalgebra. As a consequence, a complete list of the locally finite, simple-pointed Hopf algebras is obtained. Also, the graded automorphism group of a Hopf algebra on a Schurian Hopf quiver is determined, and the relation between this group and the automorphism groups of the corresponding Hopf quiver, is clarified.     

Good Gradings of Generalized Incidence Rings

This inquiry is based on both the construction of generalized incidence rings due to Gene Abrams and the construction of good group gradings of incidence algebras due to Molli Jones. We provide conditions for a generalized incidence ring to be graded isomorphic to a subring of an incidence ring over a preorder. We also extend Jones's construction to good group gradings for incidence algebras over preorders with crosscuts of length one or two.     

POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH

Let A be a superalgebra over a field of characteristic zero. In this paper we investigate the graded polynomial identities of A through the asymptotic behavior of a numerical sequence called the sequence of graded codimensions of A. Our main result says that such sequence is polynomially bounded if and only if the variety of superalgebras generated by A does not contain a list of five superalgebras consisting of a 2-dimensional algebra, the infinite dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with trivial and nontrivial gradings. Our main tool is the representation theory of the symmetric group.     

Primary decomposition over rings graded by finitely generated Abelian groups

Let A be a Noetherian ring which is graded by a finitely generated Abelian group G. In general, for G-graded modules there do not exist primary decompositions which are graded themselves. This is quite different from the case of gradings by torsion free group, for which graded primary decompositions always exists. In this paper we introduce G-primary decompositions as a natural analogue to primary decomposition for G-graded A-modules. We show the existence of G-primary decomposition and give a few characterizations analogous to Bourbaki's treatment for torsion free groups.     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

Finiteness of the strong global dimension of radical square zero algebras

The strong global dimension of a finite dimensional algebra A is the maximum of the width of indecomposable bounded differential complexes of finite dimensional projective A-modules. We prove that the strong global dimension of a finite dimensional radical square zero algebra A over an algebraically closed field is finite if and only if A is piecewise hereditary. Moreover, we discuss results concerning the finiteness of the strong global dimension of algebras and the related problem on the density of the push-down functors associated to the canonical Galois coverings of the trivial extensions of algebras by their repetitive algebras.     

<Emphasis Type="Italic">N</Emphasis>-homogeneous <Emphasis Type="Italic">C</Emphasis>*-algebras generated by idempotents

In this paper we consider n-homogeneous C*-algebras generated by idempotents. We prove that a finitely generated unital n-homogeneous (when n is greater than or equals 2) C*-algebra A can be generated by a finite set of idempotents if and only if the algebra A contains at least one nontrivial idempotent.     

Induced good gradings of structural matrix rings

Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group G, a G-grading set is a set of arrows with the property that any assignment of these arrows to elements of G uniquely determines an induced good grading. One of our main results is that a G-grading set exists for any transitive directed graph if G is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have G-grading sets for any cyclic group G of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.     

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

Koszulity and Koszul modules of dual extension algebras

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.     

Permutation Groups, a Related Algebra and a Conjecture of Cameron

We consider the permutation group algebra defined by Cameron and show that if the permutation group has no finite orbits, then no homogeneous element of degree one is a zero-divisor of the algebra. We proceed to make a conjecture which would show that the algebra is an integral domain if, in addition, the group is oligomorphic. We go on to show that this conjecture is true in certain special cases, including those of the form H Wr S and H Wr A, and show that in the oligormorphic case, the algebras corresponding to these special groups are polynomial algebras. In the H Wr A case, the algebra is related to the shuffle algebra of free Lie algebra theory.     

Hyperinvariant subspaces for some <Emphasis Type="Italic">B</Emphasis>–circular operators

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W^*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z^*z. We show that a large class of L^∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F₃). These results generalize some of what is known about the quasinilpotent DT–operator. Supported in part by NSF Grant DMS-0300336. with an Appendix by Gabriel Tucci     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A⊗F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras. Su, Y., Zhao, K., Second cohornology group of generalized Witt type Lie algebras and certain representations, submitted to publication